LIST OF FIGURES 5
2.2.3 Reducing an annular strand diagram. In the pictures, the green regions are
subject to type I moves, and the red and blue regions are each subject
to type II moves. The type II move on the red region in (a) breaks the
connected annular strand diagram into two connected components. . . . . . 34
2.3.1 A reduction II move causing a connected component to split into two (each
shaded region represents a component). . . . . . . . . . . . . . . . . . . . . 36
2.3.2 Two reduced annular strand diagrams (A
1
has been taken from [3]). . . . . 37
2.4.1 For a cutting path, the edge crossing in (a) is allowed and (b) is not allowed 40
2.4.2 Update of the cutting path for each reduction move. . . . . . . . . . . . . . 41
2.4.3 Status of a cutting path (a) after closing a strand diagram (crosses edge e
c
),
(b) after performing a type II reduction, and (c) after the annular strand
diagram is reduced. The numbers denote the order of the edges in the
cutting path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5.1 Three reduced annular strand diagrams in which A
1
and A
2
are isotopic. . 45
3.1.1 Overview of the solution algorithm for the conjugacy problem in F . . . . . 49
3.2.1 The Java model of the data structure used for the solution to the conjugacy
problem in F . Note that all the linked lists are doubly linked. . . . . . . . . 52
3.4.1 (a) Reduction I and (b) reduction II, with labeled edges and the split in-
volved in the reduction labeled v. The green vertices can be splits or merges,
they may not be distinct from each other or from the yellow vertices. . . . . 58
3.4.2 Special cases for updating cuttingPath during a type II move. Refer to (b)
in Figure 3.4.1 for edge and vertex labels. . . . . . . . . . . . . . . . . . . . 66
3.6.1 The encoding of s ∈ X when s is a free loop. φ(s) ∈ G is the planar graph
with a single vertex having a loop. . . . . . . . . . . . . . . . . . . . . . . . 68
3.6.2 Different input and output types for edges. The vertex on the left is a merge,
and the one on the right is a split . . . . . . . . . . . . . . . . . . . . . . . . 69
3.6.3 (a) shows the highlighted edge e
k
which is the lone output of a merge and
the lone input to a split. The encoding of e
k
produces the planar graph g
k
in (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.6.4 Encoding of x
0
x
0
to its corresponding planar graph. . . . . . . . . . . . . . 70
3.10.1A general structure for reduced annular strand diagrams with two vertices.
The red dashed circles show free loops. . . . . . . . . . . . . . . . . . . . . . 80
3.11.1The user interface of the application for the conjugacy problem in F . . . . . 83